As a math community, we still struggle with the balance between concept development and practice, and people argue vehemently on either side of what I believe is an entirely false dichotomy. Here are some of the ideas that I think inform the most productive way to think about concepts and practice.

I apologize at the start for my lack of references. Many of these ideas started with Craig Barton’s excellent podcast: Mr. Barton Maths Podcast. I have read Craig’s books, What Does this Look Like in the Classroom? by Carl Hendrick and Robin Macpherson, listened to talks by Robert and Elizabeth Bjork, followed Michael Pershan’s blog… all of these have contributed to my thinking, but I am not going to support my thoughts with specific citations.

This is the big idea. In terms of assigning work to students or engaging them in activities, I think all work falls into at least one of these categories, and this schema can help us be more effective in choosing when and why to use a particular kind of activity and the likely outcome.

I apologize at the start for my lack of references. Many of these ideas started with Craig Barton’s excellent podcast: Mr. Barton Maths Podcast. I have read Craig’s books, What Does this Look Like in the Classroom? by Carl Hendrick and Robin Macpherson, listened to talks by Robert and Elizabeth Bjork, followed Michael Pershan’s blog… all of these have contributed to my thinking, but I am not going to support my thoughts with specific citations.

This is the big idea. In terms of assigning work to students or engaging them in activities, I think all work falls into at least one of these categories, and this schema can help us be more effective in choosing when and why to use a particular kind of activity and the likely outcome.

2) On the left side you see “retrieval strength” and “storage strength.” There are a number of researchers who describe learning as a change in long term memory. For learning to be meaningful, it has to be stored in the brain and be retrievable at some later time beyond a matter of seconds. So for all practical purposes, all knowledge can be considered equivalent to memory, somehow stored in the brain.

Memories have two kinds of strength (again, this is a layperson’s understanding): storage strength and retrieval strength. Storage strength is the degree to which a memory is either isolated (low storage strength) or connected to other memories (higher). We all know that random facts are easier to remember if you can link them to something else, or how one person starting a story can make it easier for you to remember what happened next. We also know that what we call understanding often has to do with connections between related groups of memories (knowledge)

Memories have two kinds of strength (again, this is a layperson’s understanding): storage strength and retrieval strength. Storage strength is the degree to which a memory is either isolated (low storage strength) or connected to other memories (higher). We all know that random facts are easier to remember if you can link them to something else, or how one person starting a story can make it easier for you to remember what happened next. We also know that what we call understanding often has to do with connections between related groups of memories (knowledge)

Retrieval strength is the degree to which a memory is easy to bring to conscious awareness. Memories fade, and not through any fault of ours - the brain is designed that way. It is a way for the brain to prioritize important memories. Use it or lose it. If we remembered everything, our minds would be a cluttered mess, so seldom used memories fade. But if we try to recall a memory just as it fades away, that memory gets a little boost in its retrieval strength. And with repeated attempts at recall, the retrieval strength continues to build. (See the Ebbinghaus Forgetting Curve). This is why flashcards work, but only if you try to recall the information on the back side when you look at the front. Simply reading both sides doesn’t require any effort at recalling, and so doesn’t build retrieval strength.

Without repeated effort, retrieval strength fades, but storage strength seems to be permanent. As math teachers, this has two important implications: 1) It is not the fault of your instruction that the students can’t recall the area formula for a triangle or circle after that amazing lesson. That is how learning happens. The memory has to fade before we can strengthen it. 2) We have all had the experience (over and over and over) of our students telling us they have never seen _______ before, until we remind them that we did that yesterday… “Oh yeah, NOW I remember,” and off we go. That is the storage strength that lingers while the accessibility has dimmed. Again, supposed to happen. So we had better take that into account in our teaching.

3) Across the top, we have procedural knowledge and conceptual knowledge, and I am not sure I have seen those designations in any research the way I think of them. The symbols of mathematics are entirely arbitrary and carry no inherent meaning (I can’t think of any exceptions). The fraction bar, algebraic notation, exponents, even the numerals themselves could mean anything. I use procedural knowledge as the label for the knowledge of the symbols and the rules (or procedures or algorithms) that we learn as shortcuts to work with these symbols.

There is another kind of knowledge that I call conceptual knowledge, and I don’t know how to define it strictly. You might say it is externally observable or confirm-able knowledge, but I don’t know if that is always so. But here is the difference. Suppose I tell a student to cut a sandwich into fourths, and then cut one of the fourths into 5 equal pieces. And then I ask them to consider one of those pieces and tell me how many it would take to get a whole sandwich. We know that the student can answer that problem without knowing the procedure we teach for finding 1/4 ÷ 5, or even without using symbols beyond whole numbers. We know that the procedure that we use to answer 1/4 ÷ 5 requires additional conceptual knowledge, that dividing by 5 and multiplying by 1/5 represent the same mathematical operation. And we know that sometimes students are taught the procedure to get the answer 1/4 ÷ 5 = 1/20 without any reference to the underlying concepts.

So conceptual knowledge and procedural knowledge are distinct. And growth in one is not sufficient to produce growth in the other. As in the example above, knowing how to cut up sandwiches and find the answer may contribute to but does not produce algorithmic competence. And algorithmic competence is even less likely by itself to produce understanding of the underlying manipulations. We have to include in our instruction episodes which explicitly evolve links between the two, and I think we still really struggle with how to do that effectively.

4) So why is this important?

Our pedagogy has blind spots and the effect of those blind spots can be unintentionally handicapping the learning of our students. This framework can help us identify some of them.

Without repeated effort, retrieval strength fades, but storage strength seems to be permanent. As math teachers, this has two important implications: 1) It is not the fault of your instruction that the students can’t recall the area formula for a triangle or circle after that amazing lesson. That is how learning happens. The memory has to fade before we can strengthen it. 2) We have all had the experience (over and over and over) of our students telling us they have never seen _______ before, until we remind them that we did that yesterday… “Oh yeah, NOW I remember,” and off we go. That is the storage strength that lingers while the accessibility has dimmed. Again, supposed to happen. So we had better take that into account in our teaching.

3) Across the top, we have procedural knowledge and conceptual knowledge, and I am not sure I have seen those designations in any research the way I think of them. The symbols of mathematics are entirely arbitrary and carry no inherent meaning (I can’t think of any exceptions). The fraction bar, algebraic notation, exponents, even the numerals themselves could mean anything. I use procedural knowledge as the label for the knowledge of the symbols and the rules (or procedures or algorithms) that we learn as shortcuts to work with these symbols.

There is another kind of knowledge that I call conceptual knowledge, and I don’t know how to define it strictly. You might say it is externally observable or confirm-able knowledge, but I don’t know if that is always so. But here is the difference. Suppose I tell a student to cut a sandwich into fourths, and then cut one of the fourths into 5 equal pieces. And then I ask them to consider one of those pieces and tell me how many it would take to get a whole sandwich. We know that the student can answer that problem without knowing the procedure we teach for finding 1/4 ÷ 5, or even without using symbols beyond whole numbers. We know that the procedure that we use to answer 1/4 ÷ 5 requires additional conceptual knowledge, that dividing by 5 and multiplying by 1/5 represent the same mathematical operation. And we know that sometimes students are taught the procedure to get the answer 1/4 ÷ 5 = 1/20 without any reference to the underlying concepts.

So conceptual knowledge and procedural knowledge are distinct. And growth in one is not sufficient to produce growth in the other. As in the example above, knowing how to cut up sandwiches and find the answer may contribute to but does not produce algorithmic competence. And algorithmic competence is even less likely by itself to produce understanding of the underlying manipulations. We have to include in our instruction episodes which explicitly evolve links between the two, and I think we still really struggle with how to do that effectively.

4) So why is this important?

Our pedagogy has blind spots and the effect of those blind spots can be unintentionally handicapping the learning of our students. This framework can help us identify some of them.

**Access and agency**- One blind spot is underestimating the level of abstraction in mathematical symbols and the impact that has on learning. Which question is more accessible to more students:

“If a box that is 1/3 full weighs 8 pounds, how much does a full box weigh?” or

“What is x if 1/3x = 8?”

The symbols are meant to represent conceptual quantities and relationships, but we make it harder for both students when we try to teach the rules for the symbols without having first established the ideas they represent through more concrete means. Think about the two problems above. If we start at solving for x, who do we shut out of the conversation? If we want students to be less passive learners in math class, we have to give them a level of understanding that they can then go on to manipulate without our guidance. Using models like hanger or tape diagrams to develop the concepts of fractions or algebra before we introduce the symbols allows more (not all, but more) students to be understand and be active in the process.

We make it harder for teachers too. We have normalized some ridiculous contortions for trying to get students to understand and remember rules like flip and multiply or do the same thing to both sides. But it is like trying to explain music by explaining the rules of musical notation. Yes, being able to write music on the page may be the goal, but it would be so much easier to start by just humming a tune.

**Procedures are not enough**- In the end, we know that instruction that focuses just on the procedures leaves lots of substance behind. The formula for the area of a parallelogram does not by itself engage a student with the concepts of counting squares to measure area, multiplying as a shortcut for counting squares in a rectangular array, cutting and dissecting shapes with conservation of area, or generalizing a process with an algebraic formula. Procedures are just half of the math.

**Concepts are not enough**- Another blind spot (or misconception really) is that understanding concepts will lead to remembering and using procedures. Using a model to solve fraction problems does not translate directly to having efficient algorithms, which, I imagine, is why some teachers think “Why do twice the work?” I hope that you agree with me that the answer is “Because they learn twice the math.” But the point is we do have to do all the work. The model has to be established. The procedures have to be practiced - building retrieval takes retrieval practice. And the connections between the two have to be included explicitly in the instruction.

**Procedures need to be understood and concepts need to be remembered**- Once students are comfortable with the symbols, we can begin to study them as objects themselves, and build connections between things we know about procedures and symbols. Examples of this might be number theory and topics like Pythagorean triples or Open Middle questions about making the greatest or least fraction sum.

Concepts like multiplying by 1/3 is the same as dividing by 3 or any triangle is half of a parallelogram also benefit from retrieval practice. We would like that knowledge to be easily recalled by our students.

**Different kinds of knowledge benefit from different kinds of pedagogy**- The quadrant framework can help us to look at our own instruction and make more intentional choices about what kinds of knowledge we are developing and when. We can look at the activities we are currently doing with students and consider which quadrants they address. We may notice quadrants we are not visiting often enough with students, or identify activities that may be serving different purposes than we thought.

I will be posting some additional blogs focusing on each of the quadrants, and some resources I think address each one. Stay tuned.